. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems

A X = B

and

A^{T} X = B

are

(D_{R} A D_{C}) (D_{C}^{- 1} X) = D_{R} B

and

{(D_{R} A D_{C})}^{T} (D_{R}^{- 1} X) = D_{C} B,

respectively, where

D_{R}

and

D_{C}

are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.

When equilibration is used,

A

will be overwritten by

D_{R} A D_{C}

and

B

will be overwritten by

D_{R} B

(or

D_{C} B

when the solution of

A^{T} X = B

is sought).

Factorization

The matrix

A

, or its scaled form, is copied and factored using the

L U

decomposition

A = P L U,

where

P

is a permutation matrix,

L

is a unit lower triangular matrix, and

U

is upper triangular.

This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to nag_lapack_dgbsvx (f07bb) with the same matrix

A

Condition Number Estimation

The

L U

factorization of

A

determines whether a solution to the linear system exists. If some diagonal element of

U

is zero, then

U

is exactly singular, no solution exists and the function returns with a failure. Otherwise the factorized form of

A

is used to estimate the condition number of the matrix

A

. If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit.

Solution

The (equilibrated) system is solved for

X

(

D_{C}^{- 1} X

D_{R}^{- 1} X

) using the factored form of

A

(

D_{R} A D_{C}

Iterative Refinement

Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.

Construct Solution Matrix $X$

If equilibration was used, the matrix

X

is premultiplied by

D_{C}

(if

trans ='N'

) or

D_{R}

(if

trans ='T'

'C'

) so that it solves the original system before equilibration.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1: $fact$ – string (length ≥ 1)

Specifies whether or not the factorized form of the matrix

A

is supplied on entry, and if not, whether the matrix

A

should be equilibrated before it is factorized.

$fact ='F'$: afb and ipiv contain the factorized form of $A$ . If $equed \neq'N'$ , the matrix $A$ has been equilibrated with scaling factors given by r and c. ab, afb and ipiv are not modified.
$fact ='N'$: The matrix $A$ will be copied to afb and factorized.
$fact ='E'$: The matrix $A$ will be equilibrated if necessary, then copied to afb and factorized.

Constraint:

fact ='F'

'N'

'E'

2: $trans$ – string (length ≥ 1)

Specifies the form of the system of equations.

$trans ='N'$: $A X = B$ (No transpose).
$trans ='T'$ or $'C'$: $A^{T} X = B$ (Transpose).

Constraint:

trans ='N'

'T'

'C'

3: $kl$ – int64int32nag_int scalar

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

4: $ku$ – int64int32nag_int scalar

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

5: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The

n

n

coefficient matrix

A

The matrix is stored in rows

1

k_{l} + k_{u} + 1

, more precisely, the element

A_{i j}

must be stored in

ab (k_{u} + 1 + i - j, j) for ​ \max (1, j - k_{u}) \leq i \leq \min (n, j + k_{l}) .

See Further Comments for further details.

fact ='F'

and

equed \neq'N'

A

must have been equilibrated by the scaling factors in r and/or c.

6: $afb (ldafb :)$ – double array

The first dimension of the array afb must be at least

2 \times kl + ku + 1

The second dimension of the array afb must be at least

\max (1, n)

fact ='N'

'E'

, afb need not be set.

fact ='F'

, details of the

L U

factorization of the

n

n

band matrix

A

, as computed by nag_lapack_dgbtrf (f07bd).

The upper triangular band matrix

U

, with

k_{l} + k_{u}

superdiagonals, is stored in rows

1

k_{l} + k_{u} + 1

of the array, and the multipliers used to form the matrix

L

are stored in rows

k_{l} + k_{u} + 2

2 k_{l} + k_{u} + 1

equed \neq'N'

, afb is the factorized form of the equilibrated matrix

A

7: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

fact ='N'

'E'

, ipiv need not be set.

fact ='F'

, ipiv contains the pivot indices from the factorization

A = L U

, as computed by nag_lapack_dgbtrf (f07bd); row

i

of the matrix was interchanged with row

ipiv (i)

8: $equed$ – string (length ≥ 1)

fact ='N'

'E'

, equed need not be set.

fact ='F'

, equed must specify the form of the equilibration that was performed as follows:

if $equed ='N'$ , no equilibration;
if $equed ='R'$ , row equilibration, i.e., $A$ has been premultiplied by $D_{R}$ ;
if $equed ='C'$ , column equilibration, i.e., $A$ has been postmultiplied by $D_{C}$ ;
if $equed ='B'$ , both row and column equilibration, i.e., $A$ has been replaced by $D_{R} A D_{C}$ .

Constraint: if

fact ='F'

equed ='N'

'R'

'C'

'B'

9: $r (:)$ – double array

The dimension of the array r must be at least

\max (1, n)

fact ='N'

'E'

, r need not be set.

fact ='F'

and

equed ='R'

'B'

, r must contain the row scale factors for

A

D_{R}

; each element of r must be positive.

10: $c (:)$ – double array

The dimension of the array c must be at least

\max (1, n)

fact ='N'

'E'

, c need not be set.

fact ='F'

equed ='C'

'B'

, c must contain the column scale factors for

A

D_{C}

; each element of c must be positive.

11: $b (ldb :)$ – double array

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, nrhs_p)

The

n

r

right-hand side matrix

B

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b and the second dimension of the arrays ab, afb, ipiv, r, c.
$n$ , the number of linear equations, i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – double array

The first dimension of the array ab will be

kl + ku + 1

The second dimension of the array ab will be

\max (1, n)

fact ='F'

'N'

, or if

fact ='E'

and

equed ='N'

, ab is not modified.

equed \neq'N'

then, if no constraints are violated,

A

is scaled as follows:

if $equed ='R'$ , $A = D_{r} A$ ;
if $equed ='C'$ , $A = A D_{c}$ ;
if $equed ='B'$ , $A = D_{r} A D_{c}$ .

2: $afb (ldafb :)$ – double array

The first dimension of the array afb will be

2 \times kl + ku + 1

The second dimension of the array afb will be

\max (1, n)

fact ='F'

, afb is unchanged from entry.

Otherwise, if no constraints are violated, then if

fact ='N'

, afb returns details of the

L U

factorization of the band matrix

A

, and if

fact ='E'

, afb returns details of the

L U

factorization of the equilibrated band matrix

A

(see the description of ab for the form of the equilibrated matrix).

3: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv will be

\max (1, n)

fact ='F'

, ipiv is unchanged from entry.

Otherwise, if no constraints are violated, ipiv contains the pivot indices that define the permutation matrix

P

; at the

i

th step row

i

of the matrix was interchanged with row

ipiv (i)

ipiv (i) = i

indicates a row interchange was not required.

fact ='N'

, the pivot indices are those corresponding to the factorization

A = L U

of the original matrix

A

fact ='E'

, the pivot indices are those corresponding to the factorization of

A = L U

of the equilibrated matrix

A

4: $equed$ – string (length ≥ 1)

fact ='F'

, equed is unchanged from entry.

Otherwise, if no constraints are violated, equed specifies the form of equilibration that was performed as specified above.

5: $r (:)$ – double array

The dimension of the array r will be

\max (1, n)

fact ='F'

, r is unchanged from entry.

Otherwise, if no constraints are violated and

equed ='R'

'B'

, r contains the row scale factors for

A

D_{R}

, such that

A

is multiplied on the left by

D_{R}

; each element of r is positive.

6: $c (:)$ – double array

The dimension of the array c will be

\max (1, n)

fact ='F'

, c is unchanged from entry.

Otherwise, if no constraints are violated and

equed ='C'

'B'

, c contains the row scale factors for

A

D_{C}

; each element of c is positive.

7: $b (ldb :)$ – double array

The first dimension of the array b will be

\max (1, n)

The second dimension of the array b will be

\max (1, nrhs_p)

equed ='N'

, b is not modified.

trans ='N'

and

equed ='R'

'B'

, b stores

D_{R} B

trans ='T'

'C'

and

equed ='C'

'B'

, b stores

D_{C} B

8: $x (ldx :)$ – double array

The first dimension of the array x will be

\max (1, n)

The second dimension of the array x will be

\max (1, nrhs_p)

info = 0

n + 1

, the

n

r

solution matrix

X

to the original system of equations. Note that the arrays

A

and

B

are modified on exit if

equed \neq'N'

, and the solution to the equilibrated system is

D_{C}^{- 1} X

trans ='N'

and

equed ='C'

'B'

, or

D_{R}^{- 1} X

trans ='T'

'C'

and

equed ='R'

'B'

9: $rcond$ – double scalar

If no constraints are violated, an estimate of the reciprocal condition number of the matrix

A

(after equilibration if that is performed), computed as

rcond = 1.0 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

10: $ferr (nrhs_p)$ – double array

info = 0

n + 1

, an estimate of the forward error bound for each computed solution vector, such that

{‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖x_{j}‖}_{\infty} \leq ferr (j)

where

{\hat{x}}_{j}

is the

j

th column of the computed solution returned in the array x and

x_{j}

is the corresponding column of the exact solution

X

. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.

11: $berr (nrhs_p)$ – double array

info = 0

n + 1

, an estimate of the component-wise relative backward error of each computed solution vector

{\hat{x}}_{j}

(i.e., the smallest relative change in any element of

A

B

that makes

{\hat{x}}_{j}

an exact solution).

12: $work (\max (1, 3 \times n))$ – double array

info = 0

work (1)

contains the reciprocal pivot growth factor

\max |a_{i j}| / \max |u_{i j}|

. If

work (1)

is much less than

1

, then the stability of the

L U

factorization of the (equilibrated) matrix

A

could be poor. This also means that the solution

X

, condition estimator rcond, and forward error bound ferr could be unreliable. If the factorization fails with

info > 0 and info \leq n

work (1)

contains the reciprocal pivot growth factor for the leading info columns of

A

13: $info$ – int64int32nag_int scalar

info = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0 and info \leq n$: Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond = 0.0$ is returned.

W $info = n + 1$: $U$ is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector

b

, the computed solution

\hat{x}

is the exact solution of a perturbed system of equations

(A + E) \hat{x} = b

, where

|E| \leq c (n) ε P |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 9.3 of Higham (2002) for further details.

x

is the true solution, then the computed solution

\hat{x}

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖\hat{x}‖}_{\infty}} \leq w_{c} cond (A, \hat{x}, b)

where

cond (A, \hat{x}, b) = {‖|A^{- 1}| (|A| |\hat{x}| + |b|)‖}_{\infty} / {‖\hat{x}‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

. If

\hat{x}

is the

j

th column of

X

, then

w_{c}

is returned in

berr (j)

and a bound on

{‖x - \hat{x}‖}_{\infty} / {‖\hat{x}‖}_{\infty}

is returned in

ferr (j)

. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when

n = 6

k_{l} = 1

, and

k_{u} = 2

. Storage of the band matrix

A

in the array ab:

\begin{matrix} * & * & a_{13} & a_{24} & a_{35} & a_{46} \\ * & a_{12} & a_{23} & a_{34} & a_{45} & a_{56} \\ a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \end{matrix}

The total number of floating-point operations required to solve the equations

A X = B

depends upon the pivoting required, but if

n ≫ k_{l} + k_{u}

then it is approximately bounded by

O (n k_{l} (k_{l} + k_{u}))

for the factorization and

O (n (2 k_{l} + k_{u}) r)

for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement; see nag_lapack_dgbrfs (f07bh) for information on the floating-point operations required.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of this function is nag_lapack_zgbsvx (f07bp).

Example

This example solves the equations

A X = B,

where

A

is the band matrix

A = (\begin{array}{r} - 0.23 & 2.54 & - 3.66 & 0 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0 & 2.56 & 2.46 & 4.07 \\ 0 & 0 & - 4.78 & - 3.82 \end{array}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

Estimates for the backward errors, forward errors, condition number and pivot growth are also output, together with information on the equilibration of

A

Open in the MATLAB editor: f07bb_example

function f07bb_example


fprintf('f07bb example results\n\n');

kl = int64(1);
ku = int64(2);
ab = [ 0,      0,    -3.66, -2.13;
       0,      2.54, -2.73,  4.07;
      -0.23,   2.46,  2.46, -3.82;
      -6.98,   2.56, -4.78,  0];
b  = [ 4.42, -36.01;
      27.13, -31.67;
      -6.14,  -1.16;
      10.50, -25.82];
[n,nrhs] = size(b);

% Input parameter initialization
afb   = zeros(2*kl+ku+1, n);
ipiv  = zeros(n,1,'int64');
r     = zeros(n, 1);
c     = zeros(n, 1);
fact  = 'Equilibration';
trans = 'No transpose';
equed = 'No';

% Solve
[ab, afb, ipiv, equed, r, c, b, x, rcond, ferr, berr, work, info] = ...
  f07bb( ...
         fact, trans, kl, ku, ab, afb, ipiv, equed, r, c, b);

fprintf('Solution is x:\n');
disp(x);
fprintf('\nApproximate condition number = %9.3f\n',1/rcond);
fprintf('Approximate forward  errors  :\n');
fprintf('                               %11.1e\n',ferr);
fprintf('Approximate backward errors  :\n')
fprintf('                               %11.1e\n',berr);

f07bb example results

Solution is x:
   -2.0000    1.0000
    3.0000   -4.0000
    1.0000    7.0000
   -4.0000   -2.0000


Approximate condition number =    56.409
Approximate forward  errors  :
                                   1.6e-14
                                   1.9e-14
Approximate backward errors  :
                                   1.1e-16
                                   9.9e-17

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox